
QED bremsstrahlung: This process occurs if \(X\)  (or the decay products \(Y\) ) contains photons or electrically charged particles. Additional photons are produced via \(X_i^\pm \rightarrow X_i^\pm \gamma \)  branchings with probabilities that are enhanced for both soft and collinear photons. On the other hand, collinear photons dominate the spectra at the region \(x_\gamma \rightarrow E_\gamma /m_\chi \rightarrow 1\)  with the only requirement that the angle between the emitted photon and the parent particle is very small. QED processes may lead to the production of charged fermion-antifermion pairs in photon splittings (which are generally subleading) and the corresponding probabilities are enhanced at very low values of \(Q^2 = (p_f + p_{\bar{f}})^2\) . The rates of QED processes are governed by the effective electromagnetic fine-structure constant, \(\alpha _{\mathrm {EM}}\)  (illustrated in Fig. REF a).

QCD showers: If \(X\)  (or decay products \(Y\) ) contains coloured particles, then these states will undergo QCD showers. The modeling of the QCD showers is similar to the QED one. Here, we can have enhancement of soft and collinear emissions – in \(q\rightarrow qg\)  and \(g\rightarrow gg\)  – and of \(g\rightarrow q\bar{q}\)  at low virtualities. The main parameter governing the QCD showering is the effective value of the strong coupling constant, \(\alpha _S\)  (see Fig. REF b) evaluated at a scale proprtional to the shower evolution variable (\(p_\perp \)  in Pythia8). Further sets of universal corrections in the soft limit implies that the strong coupling should be defined in the CMW [1]} rather than the conventional \(\overline{\mathrm {MS}}\)  scheme. Furthermore, good agreement between Pythia8 and experimental measurements of \(e^+ e^- \rightarrow 3~\mathrm {jets}\)  [2]}, [3]} increases the value of \(\alpha _S(M_Z)\)  by about \(10\%\) . Perturbative uncertainty estimates can be performed by variation of the evolution of the renormalisation by a factor of 2 in each direction with respect to the nominal scale choice. The framework of the automated scale variations was recently implemented in Pythia [4]} implies a compensation of second-order terms which reestablishes the agreement with the CMW scheme. Variations of the no-universal (no-singular) components of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting kernels can be performed in this framework as it is detailed in [4]}.

Hadronisation and hadron decays: Any produced coloured particles must be confined inside colourless hadrons. This process — hadronisation — takes place at a distance scale of order the proton size \(\sim 10^{-15}\) m and in Pythia is modelled by the Lund string model; see [6]} for details. The most majority of photons are produced from the decays of neutral pions where the number and energy of these photons are strongly correlated with the predicted pion spectra. The description of this process is embedded in the fragmentation function, \(f(z)\) , which gives the probability for a hadron to take a fraction \(z \in [0,1]\)  of the remaining energy at each step of the (iterative) string fragmentation process (see Fig. REF c). The fragmentation function \(f(z)\)  cannot calculated from first principles but its form can be constrained by requirements such as causality. The general form can be written as
\(f(z,m_{\perp h}) = N \frac{(1-z)^a}{z}\exp \left(\frac{-b m_{\perp h}^2}{z}\right)~,\) 
where \(N\)  is a normalisation constant that guarantees the distribution to be normalised to unit integral, and \(m_{\perp h}= \sqrt{m_h^2 + p_{\perp h}^2}\)  is called the “transverse mass”, with \(m_h\)  the mass of the produced hadron and \(p_{\perp h}\)  its momentum transverse to the string direction, \(a\)  and \(b\)  are tunable parameters which will be denoted respectively by StringZ:aLund and StringZ:bLund. We note that the \(a\)  and \(b\)  parameters are extremely highly correlated. This makes it meaningless to assign independent \(\pm \)  uncertainties on them. To address this question, we implement an alternative parametrisation of \(f(z)\)  where \(b\)  is replaced by a \(<z>\)  which represents the average \(z\)  fraction taken by \(\rho \)  mesons.
\(\left<z_\rho \right> = \int _0^1 \mathrm {d}z \ z f(z,\left<m_{\perp \rho }\right>)~,\) 
which we solve (numerically) for \(b\)  at initialisation when the option StringZ:deriveBLund = on is selected in Pythia 8.235, using the following parameters:
\(\left<m_{\perp \rho }\right>^2 & = &m^2_\rho + 2( \mbox{\texttt {StringPT:sigma}})^2~,\\\left<z_\rho \right> & = &\mbox{\texttt {StringZ:avgZLund}}~.\) 

